Let $D^2=B_1(0) \cup_{\partial B_1(0)} B_1(0)$ denote the doubling in $\mathbb{R}^2$ i.e. the metric space which one gets after gluing two closed balls of radius $1$ with their induced metric coming from $\mathbb{R}^2$ along their boundary.
The distance between two points $p_1,p_2$ is just the euclidean distance if they are lying in the same ball. If two points $p_1,p_2$ lie in distinct balls, then their distance is given by the path a particle would travel in a circular billiard from $p_1$ to $p_2'$. Where $p_2'$ is the corresponding point of $p_2$ in the first ball.
Since elliptic billiards are a well studied subject, does there exist an explicit formula for the distance between two points $p_1,p_2$?